text: Lighthill rearranged the Navier–Stokes equations, which govern the flow of a compressible viscous fluid, into an inhomogeneous wave equation, thereby making a connection between fluid mechanics and acoustics. This is often called "Lighthill's analogy" because it presents a model for the acoustic field that is not, strictly speaking, based on the physics of flow-induced/generated noise, but rather on the analogy of how they might be represented through the governing equations of a compressible fluid.
  The first equation of interest is the conservation of mass equation, which reads
  where  and  represent the density and velocity of the fluid, which depend on space and time, and  is the substantial derivative.
  Next is the conservation of momentum equation, which is given by
  where  is the thermodynamic pressure, and  is the viscous (or traceless) part of the stress tensor from the Navier–Stokes equations.
  Now, multiplying the conservation of mass equation by  and adding it to the conservation of momentum equation gives
  Note that  is a tensor (see also tensor product). Differentiating the conservation of mass equation with respect to time, taking the divergence of the last equation and subtracting the latter from the former, we arrive at
  Subtracting , where  is the speed of sound in the medium in its equilibrium (or quiescent) state, from both sides of the last equation and rearranging it results in
  which is equivalent to
  where  is the identity tensor, and  denotes the (double) tensor contraction operator.
  The above equation is the celebrated Lighthill equation of aeroacoustics. It is a wave equation with a source term on the right-hand side, i.e. an inhomogeneous wave equation. The argument of the "double-divergence operator" on the right-hand side of last equation, i.e. , is the so-called Lighthill turbulence stress tensor for the acoustic field, and it is commonly denoted by .
  Using Einstein notation, Lighthill’s equation can be written as
  where
  and  is the Kronecker delta. Each of the acoustic source terms, i.e. terms in , may play a significant role in the generation of noise depending upon flow conditions considered.  describes unsteady convection of flow (or Reynolds' Stress, developed by Osborne Reynolds),  describes sound generated by viscosity, and  describes non-linear acoustic generation processes.
  In practice, it is customary to neglect the effects of viscosity on the fluid, i.e. one takes , because it is generally accepted that the effects of the latter on noise generation, in most situations, are orders of magnitude smaller than those due to the other terms. Lighthill provides an in-depth discussion of this matter.
  In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present.
  Finally, it is important to realize that Lighthill's equation is exact in the sense that no approximations of any kind have been made in its derivation.